FirstAudi04

I am a college student on a tie budget but before heading back for my last year I stretched my budget and bought a 2004 A4 1.8T (not Quattro). I’ve always loved Audi’s, especially the TT, and when I found this one I couldn’t resist.
Problem: While I was driving home for winter break the temp gauge started acting weird (bouncing up and down and then stuck at full hot) and then the red coolant warning light came on. I pulled over and checked the engine. No coolant leaks and the coolant reservoir fluid was between max and min. Since I only had about 20 miles to go to get home, I got back in started it up and turn the heat all the way up just to be safe. The gauge went down and the coolant warning light did not come back on. Yesterday after sitting all night in the cold (I live in Michigan) when I started it up the temp gauge bounced around again and the coolant warning light came on right away. Also I noticed that the cooling fan came on right away too. Since the outside temps were in the 20’s there’s no way it could be over heating and I don’t think the fan should have come on.
I did some reading on here and think it might be the coolant temperature sensor or the thermostat. I’m leaning toward the coolant temperature sensor because:
• There’s no loss of coolant
• The heater inside the car gets warm. I think that means the coolant is getting into the engine and heating up so I don’t think it’s the thermostat.
I called the local Audi dealer and they wanted $179 to diagnosis the problem – more than I can spend just to find out what is wrong. I found an independent shop that specializes in Audi’s that has more reasonable prices but they can’t get me in until Friday. So I have two questions:
1. I have about a 40 minute commute (one way) to a job I’m working at over the break. Is it ok to drive until Friday?
2. Any thoughts on what the problem could be? Coolant temperature sensor? Thermostat? Something else?
Thanks in advance for the help and advice!